VGAM (version 1.1-6)

# genpoisson0: Generalized Poisson Regression (Original Parameterization)

## Description

Estimation of the two-parameter generalized Poisson distribution (original parameterization).

## Usage

genpoisson0(ltheta = "loglink", llambda = "logitlink",
itheta = NULL, ilambda = NULL, imethod = c(1, 1),
ishrinkage = 0.95, glambda = ppoints(5),
parallel = FALSE, zero = "lambda")

## Arguments

ltheta, llambda

Parameter link functions for $$\theta$$ and $$\lambda$$. See Links for more choices. In theory the $$\lambda$$ parameter is allowed to be negative to handle underdispersion, however this is no longer supported, hence $$0 < \lambda < 1$$. The $$\theta$$ parameter is positive, therefore the default is the log link.

itheta, ilambda

Optional initial values for $$\lambda$$ and $$\theta$$. The default is to choose values internally.

imethod

See CommonVGAMffArguments for information. Each value is an integer 1 or 2 or 3 which specifies the initialization method for each of the parameters. If failure to converge occurs try another value and/or else specify a value for ilambda and/or itheta. The argument is recycled to length 2, and the first value corresponds to theta, etc.

ishrinkage, zero

See CommonVGAMffArguments for information.

glambda, parallel

See CommonVGAMffArguments for information. Argument glambda is similar to gsigma there and is currently used only if imethod = 1.

## Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

## Warning

Although this family function is far less fragile compared to what used to be called genpoisson() it is still a good idea to monitor convergence because equidispersion may result in numerical problems; try poissonff instead. And underdispersed data will definitely result in numerical problems and warnings; try quasipoisson instead.

## Details

The generalized Poisson distribution (GPD) was proposed by Consul and Jain (1973), and it has PMF $$f(y)=\theta(\theta+\lambda y)^{y-1} \exp(-\theta-\lambda y) / y!$$ for $$0 < \theta$$ and $$y = 0,1,2,\ldots$$. Theoretically, $$\max(-1,-\theta/m) \leq \lambda \leq 1$$ where $$m$$ $$(\geq 4)$$ is the greatest positive integer satisfying $$\theta + m\lambda > 0$$ when $$\lambda < 0$$ [and then $$Pr(Y=y) = 0$$ for $$y > m$$]. However, there are problems with a negative $$\lambda$$ such as it not being normalized, so this family function restricts $$\lambda$$ to $$(0, 1)$$.

This original parameterization is called the GP-0 by VGAM, partly because there are two other common parameterizations called the GP-1 and GP-2 (see Yang et al. (2009), genpoisson1 and genpoisson2) that are more suitable for regression. However, genpoisson() has been simplified to genpoisson0 by only handling positive parameters, hence only overdispersion relative to the Poisson is accommodated. Some of the reasons for this are described in Scollnik (1998), e.g., the probabilities do not sum to unity when lambda is negative. To simply things, VGAM 1.1-4 and later will only handle positive lambda.

An ordinary Poisson distribution corresponds to $$\lambda = 0$$. The mean (returned as the fitted values) is $$E(Y) = \theta / (1 - \lambda)$$ and the variance is $$\theta / (1 - \lambda)^3$$ so that the variance is proportional to the mean, just like the NB-1 and quasi-Poisson.

For more information see Consul and Famoye (2006) for a summary and Consul (1989) for more details.

## References

Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15, 791--799.

Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions, Boston, USA: Birkhauser.

Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall.

Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Applications. New York, USA: Marcel Dekker.

Yang, Z., Hardin, J. W., Addy, C. L. (2009). A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model. J. Statist. Plann. Infer., 139, 1514--1521.

Yee, T. W. (2020). On generalized Poisson regression. In preparation.

Genpois0, genpoisson1, genpoisson2, poissonff, negbinomial, Poisson, quasipoisson.

## Examples

Run this code
# NOT RUN {
gdata <- data.frame(x2 = runif(nn <- 500))
gdata <- transform(gdata, y1 = rgenpois0(nn, theta = exp(2 + x2),
logitlink(1, inverse = TRUE)))
gfit0 <- vglm(y1 ~ x2, genpoisson0, data = gdata, trace = TRUE)
coef(gfit0, matrix = TRUE)
summary(gfit0)
# }


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